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BROUWER VERSUS WITTGENSTEIN on the INFINITE and the LAW of EXCLUDED MIDDLE Ian RUMFITT Birkbeck College, University of London Ac

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BROUWER VERSUS WITTGENSTEIN on the INFINITE and the LAW of EXCLUDED MIDDLE Ian RUMFITT Birkbeck College, University of London Ac Grazer Philosophische Studien 89 (2014), 93–108. BROUWER VERSUS WITTGENSTEIN ON THE INFINITE AND THE LAW OF EXCLUDED MIDDLE Ian RUMFITT Birkbeck College, University of London Summary Wittgenstein and Brouwer were agreed that some of the higher mathematics of their day rested upon a projection into the in nite of methods that legitimately apply only within nite domains. In this paper I compare and assess the di er- ent treatments the two philosophers give of problematic cases involving in n- ity. For Brouwer, certain claims about in nite sequences provide exceptions to the law of excluded middle; while Wittgenstein argues that the same claims are without sense, since for him the law of excluded middle is a criterion of being a proposition. I end the paper by outlining how the intuitionist might respond to Wittgenstein’s arguments. According to Herbert Feigl, who was with him on the day, Wittgenstein was provoked into returning to philosophy by hearing L. E. J. Brouwer’s lecture, ‘Mathematik, Wissenschaft und Sprache’, in Vienna on 10 March 1928 (see the quotation from Feigl in Pitcher 1964, 8n). While Wittgen- stein’s later writings reject several central Brouwerian theses, a compari- son between these thinkers is instructive. As I hope this paper will show, Wittgenstein accepts one of Brouwer’s key negative contentions—namely, that some of the higher mathematics of their day rests upon an illegitimate projection into the in nite of methods that properly apply only within nite domains. While they di er over the remedy, agreement on that nega- tive point and Wittgenstein’s close engagement with Brouwer’s positive theory belie the widespread view—inspired by a notorious obiter dictum in the transcript of a 1939 lecture—that, for Wittgenstein, ‘Intuitionism is all bosh—entirely’ (LFM 237).1 1. e account of intuitionism that directly precedes this dictum in the lecture notes (which were taken down by some students) is in any case eccentric. 1. e intuitionists on in nity Nowadays, under the in uence of the late Sir Michael Dummett, we are apt to associate the intuitionist critique of classical mathematics and logic with the adoption of veri cationist semantic theories, in which the meaning of a declarative sentence (henceforth, a statement) is given by specifying the conditions in which a speaker would be entitled to assert it, rather than by specifying the conditions under which it would be true. It is important to set these associations aside in reading the early intuitionists, for the founding fathers of the school were not veri cationists. In a paper of 1923, Brouwer wrote that ‘a complete empirical corroboration of the inferences drawn [about the “world of perception”] is usually materially excluded a priori and there cannot be any question of even a partial cor- roboration in the case of (juridical and other) inferences about the past’ (Brouwer 1923, 336). A veri cationist would conclude from that claim that talk about the past is meaningless; Brouwer, though, expressly holds that it is meaningful. Indeed, he allows that the laws of classical logic, including Excluded Middle, may validly be applied in reasoning about the world of perception, as long as we are able to think of the ‘objects and mechanisms of [that] world … as (possibly partly unknown) nite discrete systems’ (ibid., emphasis in the original). More exactly, it is the possibility of projecting ‘a nite discrete system upon the objects in question’ that is the ‘condition of the applicability’ of Excluded Middle to judgements concerning those objects. We see here a fundamental di erence between Brouwer and Dummett. For Dummett, the basic mistake of the classical mathematicians is that they apply a realist or truth-conditional semantic theory to the language of mathematics. For Brouwer, by contrast, their error was to apply distinctively classical logical rules ‘even in the math- ematics of in nite systems’, where the rules’ condition of applicability does not obtain. A. N. Kolmogorov, another pioneer of intuitionism, agreed with Brouwer. He understood Brouwer’s writing to have ‘revealed that it is illegitimate to use the principle of excluded middle in the domain of trans nite argument’ (Kolmogorov 1925, 416). As Brouwer’s reference to ‘in nite systems’ implies, the early intuition- ists did not impugn as unintelligible expressions, such as ‘the sequence of natural numbers’, that purport to designate in nite mathematical structures. ey did, however, claim that talk about such structures, if it makes sense at all, is disguised talk about the mathematical principles that characterize them. us, to say that the natural number sequence 94.
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